A108457 -id:A108457 - cp's OEIS frontend

A051707 Number of factorizations of (n,n) into pairs (j,k).

1, 1, 1, 3, 1, 5, 1, 8, 3, 5, 1, 23, 1, 5, 5, 23, 1, 23, 1, 23, 5, 5, 1, 91, 3, 5, 8, 23, 1, 52, 1, 60, 5, 5, 5, 143, 1, 5, 5, 91, 1, 52, 1, 23, 23, 5, 1, 328, 3, 23, 5, 23, 1, 91, 5, 91, 5, 5, 1, 339, 1, 5, 23, 161, 5, 52, 1, 23, 5, 52, 1, 686, 1, 5, 23, 23, 5, 52, 1, 328, 23, 5, 1, 339, 5

Offset: 1

Yasutoshi Kohmoto

Pairs (j,k) must satisfy j>1, k>=1; (a,b)*(x,y)=(a*x,b*y); unit is (1,1).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			(6,6)=(2,1)*(3,6)=(2,6)*(3,1)=(2,2)*(3,3)=(2,3)*(3,2), so a(6)=5.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A050354, A108461, A108455, A348161 (into at most two pairs).
a(A025487) = A108460.
a(p^k) = A108457(k).
a(A002110) = A108459.
Main diagonal of A108455.

Edited by Christian G. Bower, Jun 03 2005

A091438 Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 7, 1, 4, 8, 12, 12, 11, 1, 4, 10, 16, 21, 19, 15, 1, 5, 12, 23, 31, 36, 30, 22, 1, 5, 15, 28, 45, 55, 58, 45, 30, 1, 6, 17, 37, 60, 84, 94, 92, 67, 42, 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56, 1, 7, 23, 55, 101, 161, 211, 249, 244, 211, 139, 77

Offset: 1

Wouter Meeussen and Christian G. Bower, Jan 08 2004

Number of ways to factor p^(n-k)*q^k where p and q are distinct primes and each factor is a multiple of q.

			  1;
  1, 2;
  1, 2, 3;
  1, 3, 4, 5;
  1, 3, 6, 7, 7; ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums: A000219.
Main diagonal: A000041.
a(2n,n) gives A108457.
Cf. A054225.

b:= proc(n, i, j, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i<1 or k<1, 0, `if`(j<1, b(n, i-1, i-1, k),
       b(n, i, j-1, k)+`if`(i>n or j>k, 0, b(n-i, i, j, k-j)))))
    end:
a:= (n, k)->  b(n$2, k$2):
seq(seq(a(n,k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 14 2015

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, If[k == 0, 1, 0], If[i < 1 || k < 1, 0, If[j < 1, b[n, i - 1, i - 1, k], b[n, i, j - 1, k] + If[i > n || j > k, 0, b[n - i, i, j, k - j]]]]]; a[n_, k_] :=  b[n, n, k, k]; Table[a[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz *)

G.f.: A(x,y) = Product_{i>=1, j=1..i} (1/(1-x^i*y^j)).

A108456 Table read by antidiagonals: T(n,k) = number of partitions of (n,k) into pairs (i,j) with i>0, j>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 6, 7, 7, 0, 1, 4, 8, 12, 12, 11, 0, 1, 4, 10, 16, 21, 19, 15, 0, 1, 5, 12, 23, 31, 36, 30, 22, 0, 1, 5, 15, 28, 45, 55, 58, 45, 30, 0, 1, 6, 17, 37, 60, 84, 94, 92, 67, 42, 0, 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56, 0, 1, 7, 23

Offset: 0

Christian G. Bower, Jun 03 2005

(a,b)+(x,y)=(a+x,b+y); unit is (0,0).

			1 0 0 0 0 ...
1 1 1 1 1 ...
2 2 3 3 4 ...
3 4 6 8 10 ...
5 7 12 16 23 ...
(3,2)=(2,2)+(1,0)=(2,1)+(1,1)=(2,0)+(1,2)=(1,2)+(1,0)+(1,0)=(1,1)+(1,1)+(1,0), so a(3,2)=6.

N. J. A. Sloane, Transforms

Cf. A108461, A108455. Columns 0-1: A000041, A000070. Main diagonal: A108457.

Euler transform of table whose g.f. is x/((1-x)*(1-y)).

cp's OEIS Frontend

A051707 Number of factorizations of (n,n) into pairs (j,k).

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A091438 Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.

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A108456 Table read by antidiagonals: T(n,k) = number of partitions of (n,k) into pairs (i,j) with i>0, j>=0.

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